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Probability

Question
CBSEENMA12033638

For two events A and B, let P(A) = 0.4 and P(B) = p and P(A∪ B) = 0.6
(i) Find p so that A and B are independent events.
(ii) Find p so that A and B are mutually exclusive events.

Solution
straight P left parenthesis straight A right parenthesis space equals space 0.4 comma space space space straight P left parenthesis straight B right parenthesis space equals space straight p comma space space space straight P left parenthesis straight A union straight B right parenthesis space equals space 0.6
(i) Since A  and B are independent events
 therefore space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B right parenthesis                                       ...(1)
Now    straight P left parenthesis straight A union straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis space plus space straight P left parenthesis straight B right parenthesis space minus space straight P left parenthesis straight A intersection straight B right parenthesis  
therefore space space space space space space space space space space space space space 0.6 space equals 0.4 space plus space straight p space minus space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B right parenthesis space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets
therefore space space space space space space space space space space space space space 0.6 space equals space 0.4 space plus space straight p space minus space left parenthesis 0.4 right parenthesis space straight p space space rightwards double arrow space space 0.2 space equals space 0.6 space straight p
rightwards double arrow space space space space space space space space space space space space space straight p space equals space fraction numerator 0.2 over denominator 0.6 end fraction space equals 2 over 6 space equals 1 third
(ii) Since A and B are mutually exclusive events
therefore space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 0                                                     ...(2)
Now,   straight P left parenthesis straight A space union space straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis plus straight P left parenthesis straight B right parenthesis space minus space straight P left parenthesis straight A intersection straight B right parenthesis
therefore space space space space space space space space space space 0.6 space equals space 0.4 space plus space straight p space minus space 0 space space space rightwards double arrow space space space straight p space equals space 0.2 space equals space 1 fifth

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